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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,10,13,9 X16,13,5,14 X14,7,15,8 X8,15,9,16 X2536 X4,12,1,11

Gauss Code:

{{1, -7, 2, -8}, {7, -1, 5, -6, 3, -2, 8, -3, 4, -5, 6, -4}}

Jones Polynomial:

q-11/2 - 2q-9/2 + 4q-7/2 - 6q-5/2 + 5q-3/2 - 6q-1/2 + 4q1/2 - 3q3/2 + q5/2

A2 (sl(3)) Invariant:

- q-18 - q-16 - 2q-12 + q-10 + 2q-8 + 2q-6 + 4q-4 + q-2 + 3 + q6 - q8

HOMFLY-PT Polynomial:

a-1z + a-1z3 - 2az-1 - 4az - 3az3 - az5 + 3a3z-1 + 4a3z + 2a3z3 - a5z-1 - a5z

Kauffman Polynomial:

a-2z2 - a-2z4 - 2a-1z + 5a-1z3 - 3a-1z5 + z2 + 3z4 - 3z6 + 2az-1 - 7az + 11az3 - 5az5 - az7 - 3a2 + 2a2z2 + 5a2z4 - 5a2z6 + 3a3z-1 - 7a3z + 9a3z3 - 4a3z5 - a3z7 - 3a4 + 4a4z2 - 2a4z6 + a5z-1 - 2a5z + 3a5z3 - 2a5z5 - a6 + 2a6z2 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                 1 j = 4               2   j = 2             2 1   j = 0           4 2     j = -2         3 4       j = -4       3 2         j = -6     1 3           j = -8   1 3             j = -10   1               j = -12 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[8, Alternating, 4]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[8, Alternating, 4]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 10, 13, 9], X[16, 13, 5, 14], > X[14, 7, 15, 8], X[8, 15, 9, 16], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 2, -8}, {7, -1, 5, -6, 3, -2, 8, -3, 4, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 4 6 5 6 3/2 5/2 q - ---- + ---- - ---- + ---- - ------- + 4 Sqrt[q] - 3 q + q 9/2 7/2 5/2 3/2 Sqrt[q] q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 2 -10 2 2 4 -2 6 8 3 - q - q - --- + q + -- + -- + -- + q + q - q 12 8 6 4 q q q q
In[8]:=	HOMFLYPT[Link[8, Alternating, 4]][a, z]
Out[8]=	3 5 3 -2 a 3 a a z 3 5 z 3 3 3 5 ---- + ---- - -- + - - 4 a z + 4 a z - a z + -- - 3 a z + 2 a z - a z z z z a a
In[9]:=	Kauffman[Link[8, Alternating, 4]][a, z]
Out[9]=	3 5 2 2 4 6 2 a 3 a a 2 z 3 5 2 z -3 a - 3 a - a + --- + ---- + -- - --- - 7 a z - 7 a z - 2 a z + z + -- + z z z a 2 a 3 2 2 4 2 6 2 5 z 3 3 3 5 3 4 > 2 a z + 4 a z + 2 a z + ---- + 11 a z + 9 a z + 3 a z + 3 z - a 4 5 z 2 4 6 4 3 z 5 3 5 5 5 6 2 6 > -- + 5 a z - a z - ---- - 5 a z - 4 a z - 2 a z - 3 z - 5 a z - 2 a a 4 6 7 3 7 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 1 3 1 3 3 2 4 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 3 2 2 2 4 2 6 3 > ---- + 2 t + 2 q t + q t + 2 q t + q t 2 q t

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